Generalized balances in Sturmian words

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Abstract

One of the numerous characterizations of Sturmian words is based on the notion of balance. An infinite word x on the {0,1} alphabet is balanced if, given two factors of x,w and w′, having the same length, the difference between the number of 0's in w (denoted by |w|0) and the number of 0's in w′ is at most 1, i.e. ||w|0−|w′|0|⩽1. It is well known that an aperiodic word is Sturmian if and only if it is balanced.

In this paper, the balance notion is generalized by considering the number of occurrences of a word u in w (denoted by |w|u) and w′. The following is obtained.

Theorem. Let x be a Sturmian word. Let u, w and wbe three factors of x. Then,|w|=|w′|⇒||w|u−|w′|u|⩽|u|.

Another balance property, called equilibrium, is also given. This notion permits us to give a new characterization of Sturmian words. The main techniques used in the proofs are word graphs and return words.

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